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Circuit Elements And Connections Inspired By Neurons John Robert Burger, Emeritus Professor, ECE Department, California State University Northridge
jrburger1@gmail.com
Abstract -- A new direction is struck with an analysis of natural pulses propagating along a realistic model of
a neural membrane. Described below are interesting neural pulses that support the efficient generation of a
wide variety of dense combinational and sequential logic. Although not new, the implied circuit elements and
connections are interesting and may foreshadow the future of circuit engineering.
Introduction Circuits engineers generally work with solid state connections in one form or another. The work
described below runs against this trend in that it focuses on models of natural circuitry as
accomplished by biological neurons. Neural circuits, even realistic ones are not new. But from a
scientific point of view, everyday humble neurons hold interest because they have complexity
and abilities that range far beyond the transistor. Engineers could learn much from them.
If you are a neuroscientist, you see neurons as mainly passing molecules and ions back and forth.
But for circuit enthusiasts, it seems clear that neurons are in fact electrical devices. There is
nothing new in the idea that neurons generate logic and signal each other with long tentacles
analogous to electrical wires. But it must be kept in mind that they rely on voltage pulses, and so
are very different from level sensitive logic.
Biological circuits and circuit elements are not new, nor are they of much interest for financial
gain, but circuits of this magnitude are intrinsically interesting in their own right. However, the
viewpoint is new, which is electrical as opposed to chemical.
Picturesque neural circuit models with computer analysis began to appear in mid twentieth
century [1]. This early work is now settled science, promulgated in all major neuroscience texts.
Its wide publication makes it easy and fun to explore idiosyncrasies and obscure implications of
the modeling. Some implications of interest to electrical scientists have been ignored by
neuroscientists, who are mainly interested in molecular biology and biochemistry, not circuitry.
Electrical Analysis Of A Neuron Neurons are elongated biological cells with micrometer dimensions, enclosed in a thin 1-5 nm
membrane. The membrane is not uniform in that it is spotted with larger transmembrane
molecules, roughly 1015 molecules/cm2. This membrane, with its landscape of transmembrane
molecules, in conjunction with outside ions such as sodium, and inside ions such as potassium, is
electrically active.
In equilibrium, there is an interior potential of about −70 mV relative to the outside, because of
natural exchanges of charges between the inside and outside ions. This potential results in a high
field across the membrane, roughly 140 kV/cm. Lightening in air requires only about 30 kV/cm.
As a result, while in equilibrium, the molecules are held quite tightly.
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But if an external force reduces the voltage across the membrane from −70 mV to about −55
mV, the molecules become slightly relaxed, and permit a passage of positive charge to the inside
which builds up locally. This further triggers the membrane so that a rising pulse is initiated.
Figure 1 illustrates a membrane that is embedded with two larger molecules; the top represents
the outside where sodium ions (Na+) are chaotically undergoing thermal agitation according to a
probability distribution. The bottom represents the inside where potassium (K+) ions are
similarly in motion.
Figure 1 Membrane close-up with idealized ion channels
The larger embedded molecules represent ion channels. Once channels in the membrane are
triggered, sodium ions will arrive first, mainly since they are smaller and move faster. The
fastest of them may actually plough through the channel to directly capture an electron inside the
region. When this happens, the interior contains incorrect atoms, and ions, which must be
pumped out later, a process that consumes calories.
The vast majority in a Maxwell-Boltzmann distribution are moving at relatively low speeds, and
are expected to meander near (or into the mouth of) a channel at a lower speed. Electrons inside
are free briefly, and tend to flood everywhere; a passing sodium ions may capture one, thus
contributing to positive charge inside. The net result is a rising voltage inside the neuron.
Potassium ions also approach the ion channels but are slightly slower in speed. They arrive at a
lower rate, and take over fewer channels. The fastest of them may escape the neuron, although
the lost potassium must be replaced, which dissipates calories. The vast majority move slowly
near the region of the channel, and might capture an electron from the outside. Actions like this
bring negative charge into the region and slow the rise of the voltage pulse.
Positive pulse buildup naturally terminates as positive voltage builds up to a peak of about +40
mV. Near this point a strong reversing field tends to trap ions inside the channel, which are held
with a time constant, and they repel other ions away from the channel. Meanwhile, the
potassium channels keep working, causing the pulse to be reduced in voltage. Eventually it
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undershoots to about −90 mV, which is below its rest voltage of −70 mV. The associated
electrical stress on the channel molecules causes them to reset, thus closing all ion channels. In
due course, voltage drifts back to equilibrium at −70 mV. Re-triggering is possible.
A neural pulse is surely an amazement, and it is an entirely natural process; ideal pulses are
relatively efficient since ionic conditions are restored, not like transistors, in which charge is
dissipated in stray resistance during switching, necessitating extra cooling.
Propagation Of A Neural Pulse Electricity in metallic conductors travels with a time delay near the speed of light (3x108
mm/ms); neural pulses propagate very much slower however, roughly 1 mm/ms. It will be seen
that this is still adequate for small neurons in a biological brain. To fully understand neural
pulses, we consider a micron-sized model of a realistic neuron: A slender tube analogous to a
dendrite is surrounded by a thin membrane with dielectric properties as in a neuron. Short
lengths of dendrite can be modeled as in Figure 2 with many segments. Segments could be any
convenient length in a scalable model, but here each segment is assigned a fixed diameter D = 1
x 10-4 cm, and length L = 0.1 cm.
Figure 2 Model of a dendritic path
Consider a single segment. To facilitate a computerized analysis, realistic parameters are
assumed as in Table 1. These parameters are only estimations, but provide a reasonable neural
pulse.
Table 1 Physical Model
Rest Potential
Trigger Voltage (both Na and K currents)
Na Current Cutoff Voltage
K Current Cutoff Voltage
Sodium Current
Potassium Current
Membrane Capacitance
Membrane Conductance
Internal Resistivity ρ
−70 mV
−55 mV
+40 mV
−90 mV
67.25uA/cm2
30.4 uA/cm2
1 uF/cm2
0.3 mS/cm2
15.7 Ω-cm
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Table 2 summarizes equations for computing electrical parameters from physical parameters.
Table 2 Equations
L = 1 mm = 0.1 cm;
D = 1 u = 1.0 e-4 cm;
ACS = π (D/2)2;
R1 = ρL/ACS =15.7 L/ACS Ω;
ASIDE = π DL;
C1 = 1uF/cm2 ASIDE;
RLOSS1 = 1/(0.3 e-3 ASIDE) Ω;
INa1 = 0. 06725 mA/cm2 ASIDE;
IK = 0.0304 mA/cm2 ASIDE
The given segment has the circuit model shown in Figure 4. Series resistance R1 is high, roughly
200 M because the low cross sectional area. Membrane loss RLOSS1 is fairly high, about 106 M,
because a membrane is basically extremely thin and is slightly conductive Table 3 lists the
parameters used in this model
Table 3 Simulation Parameters
R1 = 200 M
C1 = 31.4 pF
RLOSS1 = 106 M
INa1 = 2.11 nA
IK1 = 0.955 nA
Figure 4 Circuit model of a segment
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This model is piecewise linear in that sodium current is triggered to begin when nodal voltage is
−55 mV, but is cut off when voltage reaches +40 mV. Potassium current is also triggered at −55
mV but is not cut until voltage falls back to −90 mV. These properties were simulated using
controlled voltage sources in WinSpice to create hysteresis loops, the outputs of which drive
inputs to controlled current sources [2].
Under this model a pulse appears as in Figure 5. This is reasonably close to what is physically
observed experimentally. A pulse in segment 1 soon triggers a pulse in segment 2 when
membrane potential reaches −55 mV. Pulses propagate by triggering new identical pulses in
resting adjacent segments, appearing to propagate without attenuation or dispersion. After about
15 ms in this model a pulse appears in segment n = 10.
Figure 5 Calculated neural pulse and its propagation
There are several interesting features of such neural pulses. For instance, it is found by
simulation that when a pulse reaches an unterminated end of a dendrite, it disappears. But if it
encounters a capacitive load of sufficient magnitude, as it would at the central body of a neuron,
it is reflected, giving a back propagation. Back propagating pulses sometimes encounter an
approaching pulse, and when they collide, both pulses are annihilated. This cuts the rate of
arriving pulses in half.
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Boolean Logic In Neurons Neural logic has been pondered for a long time [3 - 11]. Exposed dendrites as described above
typically form a large tree with many branches and intersections. The trunk attaches to an active
termination which is a slightly larger body, usually a main body known as the soma, whose
membrane is also active. The neuron's components are illustrated in Figure 6. Stemming from
the main body is a single insulated conductor, known as an axon, that can branch out to remote
locations.
Pulses in axons propagate via ordinary conduction, but with approximately 2 um nodes of
exposed membrane roughly every millimeter to refresh the pulse shapes. The conductive lengths
are insulated with a natural growth called myelination that reduces membrane capacitance; this
in turn increases the speed of the conducted pulses. The tips of an axon, or buttons, are capable
of making connections to other neurons.
Figure 6 Components of a neuron
Dendrites can be very extensive with thousands of branches with each intersection potentially a
logic gate, where true is defined to be pulsating, false is defined to be rest, that is, no pulses.
Probably the most basic type of Boolean logic is the OR function, which is generated whenever
dendrites merge into a trunk of a dendritic tree as in Figure 7. In this intersection pulses will be
forwarded on one or both inputs. Normally the OR function produces back propagations when
one input is sending pulses while the other is not, as suggested in the figure.
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Figure 7 Merging branches of dendrites
Simulations have indicated that under certain conditions the OR becomes an exclusive OR,
usually termed an XOR, which is an especially important possibility since it implies a NOT
function. This results for one input while assuming that the other input is held true, or activated
with pulses. The XOR is generated by virtue of the fact that if two pulses arrive simultaneously,
they annihilate each other. Arriving simultaneously is an important condition.
To force arriving pulses to interact to give an XOR, there must be a slight reduction in local
membrane capacitance at the junction but not necessarily a corresponding reduction in charging
current. Physically this can be accomplished in various ways, for instance, by having a partial
insulation of the membrane at the junction that is balanced by a subtle change in membrane
properties to maintain charging currents. Figure 8 attempts to illustrate an XOR gate by showing
a junction with reduced surface area, and hence, lower local capacitance.
Figure 8 Branches adjusted to give XOR
Figure 9 shows a simulation of an XOR with only one input active. Note that there is an output
as well as a back propagation on the other input. Figure 10 indicates that two inputs arriving
together will produce no output whatsoever, as expected of an XOR gate.
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Figure 9 XOR with one input (Input at A; output at Y; back propagation at B)
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Figure 10 XOR with two inputs, showing no output
The possibility of a dendritic AND gate is fairly obvious and is illustrated in Figure 11. A region
of the membrane is passivated with insulation. The insulation is such that one pulse cannot
trigger an output, whereas two pulses arriving together will provide enough charge through the
passive region to trigger an output pulse.
Figure 11 Dendritic AND gate
In summary, dendritic logic can involve thousands of gates per neuron. However, for the XOR
gates, and also for AND gates, pulses must arrive simultaneously.
In contrast to dendritic logic, another form of logic occurs in nodes such as the main body,
similar to what is assumed in traditional artificial neural networks. It is termed enabled logic
because it is easy to have an enabling input that must be active with pulses to wake the system.
Enabled OR-AND gates are fairly obvious. The AND, for example, merely accumulates charge
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in a terminating load capacitance, and if two or more inputs are providing pulses, enough voltage
builds up to trigger the load.
The NOT gate necessitates inhibitory synapses as introduced below.
Enabled logic is not as sensitive to the timing of pulses. However, it tends to be limited to one
logic gate per neuron, usually at the main body, so it's logical promise is less than that of
dendritic logic.
Excitatory And Inhibitory Synapses Triggering is also possible using larger excitatory ions that are able to approach close to the
membrane, thus upsetting transmembrane molecules. They are permitted to approach closely
with the aid of an appropriate synapse whose chief purpose is to deliver excitatory ions from the
tips of an axon of another neuron. Synapses never touch, exactly, but involve a small gap of
perhaps 20 nm. Eventually most of the excitatory ions within a synapse are attracted back to the
presynaptic tip when the tip goes negative, as it will do when it returns to rest. This terminates a
pulse burst.
Excitatory synapses can be modeled as voltage controlled current sources. Such models inject
charge into a region of membrane capacitance, thus triggering it.
Inhibitory ions can also be delivered by appropriate synapses. Inhibitory ions are usually applied
somewhere nearer the soma. They tend to stop pulses from passing through, and in this sense
they act like membrane insulation controllable by neural signals. Inhibitory synapses have been
modeled as voltage-controlled current sources that inject negative charge (that is, they remove
positive charge from a region). Removing positive charge from a region inside a membrane
tends to prevent triggering.
Without the availability of negative weighting factors as commonly assumed in artificial
neurons, one wonders if a NOT gate is even possible in a realistic neuron. An enabled NOT
requires a synapse that can be commanded to release inhibitory ions that stop certain signals
from propagating. Figure 12 suggests a NOT gate.
Figure 12 Enabled NOT gate
Assume that a pulse burst is applied to the b input and is held there for the duration of the logic
event. An input signal applied in the same time frame at the point labeled a serves release fast
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acting inhibitory ions. The inhibitory ions will stop propagation of pulses from b. This means
there will be no output at point z.
Removing the signal at point a will remove the inhibitory ions and will permit pulses from b to
trigger the soma and to produce an output. In a time frame of interest during which a and b are
active, a NOT gate results: z = a'. It may be noted that enabled NOT gates, and also those based
on the dendritic XOR typically require pulses to be applied to an input during the time frame of
the logic event.
These gates are termed enabled because basal dendrites can precharge the soma to enable it to be
triggered with regular pulses. On the other hand, if the soma is not precharged, it can remain
asleep and inactive even though there are regular pulses being applied.
The enabled XOR is possible using additional neurons.
Sequential Logic Moving beyond combinational logic, a neuron whose output synapses back to its own input is
capable of cycling a pulse indefinitely, and therefore is a novel sort of latch that has been studied
using simulation [2]. Ideally a single pulse is circulated. A single pulse comes from a physically
distinctive weak synapse, or alternatively, with a special circuit that converts a pulse burst into a
single pulse.
Resettable latches are not logically reversible. However, a single neuron that feeds back on itself
can serve as a controlled toggle, using combinations of excitatory and inhibitory synapses.
Combinational logic also could be used to implement a toggle, but at the expense of extra
neurons. Toggles are logically reversible and are potentially energy-efficient.
Figure 13 shows one of several possibilities for neural toggling, this one using combinations of
excitatory and inhibitory synapses. This method was verified with computerized simulations of
neurons, and is perhaps the simplest in that it uses only one neuron. The plan is to excite a
circulating pulse at a selected point in the loop, and simultaneously to inhibit pulse propagation
at a different (well chosen) point in the loop. Synapses are represented by little amplifiers
(triangles). Synapse S1, a weak excitatory synapse, serves to inject a pulse into the loop.
Synapse S2, a weak excitatory synapse serves to close the loop, creating a single circulating
pulse that propagates clockwise. The blocks labeled Delay1, 2, 3 and 4 represent the natural
delay of a given dendritic path.
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Figure 13 The neuron as a toggle circuit
Synapse S3 is a fast inhibitory synapse modeled with negative charge injection. A trigger is
applied simultaneously to both synapses S1 and S3 in order to toggle from no pulses to regular
pulses at the OUT point, or from regular pulses to no pulses. Simulation in a computer indicates
that negative charge injection by S3 has little effect on a pulse initiated by S1, probably because
the region near S3 is already charged negatively and for practical purposes is at rest when the
circulating pulse arrives.
However, when a pulse is propagating, another trigger to D3 will indeed stop the cycling.
Circulation stops because charge is drained from a wide region surrounding S3, taking necessary
charge away from the rather delicate circulating pulse, thus terminating pulse propagation.
The above circuit uses only a single neuron: A trigger begins an output pulse burst if the circuit
is at rest; if a pulse is circulating it stops it. There are other interesting and effective ways to get
a neural toggle circuit but this one is the simplest. Note that this circuit is easily converted into a
set-clear latch, if needed; also that the rate of pulse cycling can be reduced, if desired, by
changing the length of the delay path, or alternately by using ions to modify delay properties.
Controlled Toggling A controlled toggle is one that toggles if and only if one or more control signals are true. A
controlled toggle may be implemented with the aid of a neural AND gate as in Figure 14.
Figure 14 Creating a controlled toggle
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The individual pulses to the AND gate could be synchronized, but assuming they are not, the
AND neuron would have to be an enabled type. Controlled toggles possess computational
possibilities, and may be important to mental processing. Controlled toggling has capability for
massively parallel computing, including the prioritization of mental images and recalls.
Reversibility and power dissipation in toggles are well-studied advantages [12,13].
Detailed memory theory in neurons is beyond the scope of this article. Nonetheless, it may be
noted that short term memory is available in a neuron in an obvious way; recall that the actions
of the potassium ions inside a dendrite served to reduce a pulse; so by inhibiting the actions of
potassium, a longer positive pulse theoretically results, perhaps a few seconds. This longer
dendritic pulse can serve to continuously trigger an extended pulse burst at a normal soma,
giving nearly regular pulses that signal a short term memory.
Long term memory is available in a single recursive neuron as mentioned above, although a
recursive circuit might require a constant trickle of energy. Researchers have proposed that long
term potentiation (LTP) gives long term memory [14]. Under this mechanism a LTP synapse is
subjected to a strong burst of pulses, instilling extra charge. The existence of extra charge, and
therefore extra voltage, can be detected as in Figure 15.
Figure 15 Detecting the presence of potentiation
LTP receptors are denoted by a capacitor symbol, into which a special charge is assumed to be
instilled [10]. Subsequently a READ signal in the form of a single pulse R from a weak synapse,
is enough to cause a pulse burst at the output Q. Without LTP a single pulse would not be
enough to trigger a neural pulse, giving no output at Q. This form of memory would require two
neurons, one for an input OR gate, and one with LTP synapses for an output neuron.
An overview for non experts of the above elements and circuits and how they might support
brain operations is now available [11].
Conclusions Molecular biology is exceedingly popular, but provides few clues about a brain's computational system.
Beyond molecules, a brain is obviously a working complexity of circuits. There is no reason to suspect
that a brain violates even the smallest aspect of circuit theory or, for that matter, any known principle of
physics. So when explaining a brain, circuits and systems theory is, without question most relevant.
There is nothing new in this article except the point of view, which is electrical instead of
chemical. This article has explained how neural pulses assume a standard shape, and are
propagated along an exposed membrane. In reality neurons are quite complex, but for clarity,
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simple models were used to explain dendritic logic, which occurs at a specialized junction of
dendritic tubes. Dendritic junctions were shown to be capable of OR, AND, XOR and NOT
gates. Thousands of such gates per neuron are possible. However, for dendritic AND, XOR and
NOT logic, pulses must arrive at roughly the same time
Another form of neural logic has been termed enabled logic, and is more akin to what is used in
traditional artificial neural networks. As a simple example, the body of a neuron has a larger
capacitance that can be precharged to enable a variety of logic. Output occurs only if the
accumulated voltage, as built up by arriving voltage pulses, reaches a triggering threshold.
Logical possibilities include AND, OR, and NOT gate. Assuming logic only at the body there is
merely one gate per neuron.
This article introduced sequential logic based on the controlled toggle, which is a single neuron
whose output is either a stream of pulses, or no pulses. Controlled toggles change state only if
their control signals are true, or pulsating. Toggles are modeled above as being implemented
with excitatory and inhibitory ions, although there are other ways to accomplish toggling. Large
systems of controlled toggles are known to be capable of massively parallel arithmetic and
generalized computations, and would aid very nicely in brain operations.
Neural circuit elements as above can support large blocks of logic, as well as complex sequential
circuits; also available are short-term memory neurons, and large associative memories with
long-term capability, as in a brain. All of these natural wonders are important to engineers who
might learn from the human brain, famous for ample computing with low power dissipation. For
example, a human brain uses only about 6 watts, roughly that of a dim light, and apparently does
so without regard to the level of mental effort.
The purpose of this work is not to simulate neural behavior, since this has been covered years
ago. Rather the purpose is to learn how to manufacture neurons and neural systems as good as
the biological ideal. To facilitate this, one must understand neurons better. A crucial question is,
why does a brain have such good computational abilities with relatively low power dissipation?
There are many possibilities, but to spur thought, the following are suggested:
1) High speed pulses are nonexistent in a brain, where rise times are typically beyond a
millisecond. Thus charging, in an axon for instance, occurs with relatively low currents and
therefore low power dissipations in series resistance. Although they are slow, logic is abundant
because of a massively parallel system involving millions of neurons.
2) It has been conjectured that neural pulses are physically reversible in the sense that they derive
energy from relevant ionic solutions (and not oxygenated hemoglobin); as the pulse subsides,
charge balance is restored, and so energy is partially returned, something like a capacitor
discharging into an inductance that in turn recharges the capacitor. Unfortunately, biology is less
than ideal in that foreign ions may enter into a neuron and need to be pumped out; lost internal
ions also need to be re-manufactured. Moreover, there are losses in the neural membrane as it
sustains its rest potential, which uses calories.
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3) Assuming a brain is efficient, as nature generally is, it will employ toggle neurons (the scope
of toggle logic in a biological brain is unknown). Toggle logic is logically reversible, and this
implies low energy usage. Ordinary latches, when cleared, lose their information while also
dissipating capacitive energy in stray resistance. But a controlled toggle system need not be
cleared and can be logically reversible.
Finally, this article views neurons as electrical devices that undergo fairly efficient processes.
The analysis above is a first step toward shedding light on what has been a shadowy area
dominated by biologists and neuroscientists who usually do not focus on electrical circuits.
==
References [1] A L Hodgkin and A F Huxley, "A quantitative description of membrane current and its
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[2] J R Burger, Brain Theory From A Circuits And Systems Perspective -- How Electrical
Science Explains Neuro-circuits, Neuro-systems, and Qubits, New York: Springer, 2013,
Appendix A.
[3] P Fromherz, V Gaede, "Exclusive-OR function of a single arborized neuron." Biol. Cybern.
69, Aug. 1993, pp. 337.
[4] BW Mel, "Information Processing in Dendritic Trees." Neural Comp. 6, 1994, pp. 10311085.
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[8] J R Burger, "XOR at a Single Point where Dendrites Merge." arXiv:1004.2280 [cs.NE]
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[9] J R Burger, "Neural Networks that Emulate Qubits." NeuroQuantology 9, 2011, pp. 910916.
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[11] J R Burger, The Synthesis of a Neural System to Explain Consciousness: Neural Circuits,
Neural Systems and Wakefulness for Non-Specialists. Eugene, OR: Luminare Press, 2014.
[12] E Fredkin, T Toffoli, "Conservative logic." International Journal of Theoretical Physics
(Springer Netherlands) 21 (3), April 1982, pp. 219–253.
[13] J R Burger, Human Memory Modeled with Standard Analog and Digital Circuits:
Inspiration for Man-made Computers. Hoboken, NJ, Wiley, 2009.
[14] T Bliss, G L Collingridge, RGM Morris, Long-Term Potentiation: Enhancing Neuroscience
for 30 Years. Oxford University Press 2004.
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